2.32   ODE No. 32

\[ y'(x)+y(x)^2 \sin (x)-2 \tan (x) \sec (x)=0 \]

✓ Mathematica : cpu = 0.293864 (sec), leaf count = 34

DSolve[-2*Sec[x]*Tan[x] + Sin[x]*y[x]^2 + Derivative[1][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \frac {\csc (x) (-2 \sin (x) \cos (x)+c_1 \tan (x) \sec (x))}{\cos ^2(x)+c_1 \sec (x)}\right \}\right \}\]

✓ Maple : cpu = 0.206 (sec), leaf count = 28

dsolve(diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0,y(x))
 

\[y \left (x \right ) = \frac {-2 \cos \left (x \right )^{3} c_{1} -2}{\left (\cos \left (x \right )^{3} c_{1} -2\right ) \cos \left (x \right )}\]

Hand solution

\begin{align} y^{\prime }+y^{2}\sin \left ( x\right ) -2\frac {\sin x}{\cos ^{2}x} & =0\nonumber \\ y^{\prime } & =2\frac {\sin x}{\cos ^{2}x}-y^{2}\sin \left ( x\right ) \nonumber \\ & =P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\tag {1}\end{align}

This is Ricatti first order non-linear ODE. \(P\left ( x\right ) =2\frac {\sin x}{\cos ^{2}x},Q\left ( x\right ) =0,R\left ( x\right ) =-\sin \left ( x\right ) \). A particular solution is \(y_{p}=\frac {1}{\cos x}\), therefore the solution is

\begin{align*} y & =y_{p}+\frac {1}{u}\\ y & =\frac {1}{\cos x}+\frac {1}{u}\end{align*}

Hence

\[ y^{\prime }=\frac {\sin x}{\cos ^{2}x}-\frac {u^{\prime }}{u^{2}}\]

Equating this to RHS of (1) gives

\begin{align*} \frac {\sin x}{\cos ^{2}x}-\frac {u^{\prime }}{u^{2}} & =2\frac {\sin x}{\cos ^{2}x}-y^{2}\sin \left ( x\right ) \\ & =2\frac {\sin x}{\cos ^{2}x}-\left ( \frac {1}{\cos x}+\frac {1}{u}\right ) ^{2}\sin \left ( x\right ) \\ & =2\frac {\sin x}{\cos ^{2}x}-\left ( \frac {1}{\cos ^{2}x}+\frac {1}{u^{2}}+\frac {2}{u\cos x}\right ) \sin \left ( x\right ) \end{align*}

Hence

\begin{align*} -\frac {u^{\prime }}{u^{2}} & =-\frac {\sin x}{\cos ^{2}x}+2\frac {\sin x}{\cos ^{2}x}-\frac {\sin \left ( x\right ) }{\cos ^{2}x}-\frac {\sin \left ( x\right ) }{u^{2}}-\frac {2\sin \left ( x\right ) }{u\cos x}\\ & =-\frac {\sin \left ( x\right ) }{u^{2}}-\frac {2\sin \left ( x\right ) }{u\cos x}\end{align*}

Or

\begin{align*} u^{\prime } & =\sin \left ( x\right ) +\frac {2u\sin \left ( x\right ) }{\cos x}\\ u^{\prime }-2u\tan \left ( x\right ) & =\sin \left ( x\right ) \end{align*}

Integrating factor is \(e^{-2\int \tan xdx}=e^{2\ln \left ( \cos x\right ) }=\cos ^{2}\left ( x\right ) \). Hence the above becomes

\[ d\left ( u\cos ^{2}x\right ) =\cos ^{2}\left ( x\right ) \sin \left ( x\right ) \]

Integrating both sides

\begin{align*} u\cos ^{2}x & =\int \cos ^{2}\left ( x\right ) \sin \left ( x\right ) dx+C\\ & =\frac {-1}{3}\cos ^{3}\left ( x\right ) +C \end{align*}

Hence

\[ u=\frac {-1}{3}\cos \left ( x\right ) +\frac {C}{\cos ^{2}x}\]

Therefore

\begin{align*} y & =y_{p}+\frac {1}{u}\\ & =\frac {1}{\cos x}+\frac {1}{\frac {-1}{3}\cos \left ( x\right ) +\frac {C}{\cos ^{2}x}}\\ & =\frac {1}{\cos x}+\frac {3\cos ^{2}x}{3C-\cos ^{3}\left ( x\right ) }\end{align*}

Let \(3C=C_{1}\)

\[ y=\frac {1}{\cos x}+\frac {3\cos ^{2}x}{C_{2}-\cos ^{3}\left ( x\right ) }\]

Verification

restart; 
ode:=diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0; 
my_sol:=1/cos(x)+ 3*cos(x)^2/(_C1-cos(x)^3); 
odetest(y(x)=my_sol,ode); 
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